Figuring for Yourself

Suppose the Hubble constant were not 22 but 33 km/s per million light-years. Then what would the critical density be?

Assume that the average galaxy contains 10¹¹ M_Sun and that the average distance between galaxies is 10 million light-years. Calculate the average density of matter (mass per unit volume) in galaxies. What fraction is this of the critical density we calculated in the chapter?

The CMB contains roughly 400 million photons per m³. The energy of each photon depends on its wavelength. Calculate the typical wavelength of a CMB photon. Hint: The CMB is blackbody radiation at a temperature of 2.73 K. According to Wien’s law, the peak wave length in nanometers is given by ${\text{λ}}_{\text{max}}=\frac{3\times {10}^6}{T}.$ Calculate the wavelength at which the CMB is a maximum and, to make the units consistent, convert this wavelength from nanometers to meters.

Following up on Exercise 29.25 calculate the energy of a typical photon. Assume for this approximate calculation that each photon has the wavelength calculated in Exercise 29.25. The energy of a photon is given by $E=\frac{hc}{\text{λ}},$ where h is Planck’s constant and is equal to $6.626\times {10}^{-34}\text{ J}\times \text{s},c$ is the speed of light in m/s, and λ is the wavelength in m.

Continuing the thinking in Exercise 29.25 and Exercise 29.26, calculate the energy in a cubic meter of space, multiply the energy per photon calculated in Exercise 29.26 by the number of photons per cubic meter given above.

Continuing the thinking in the last three exercises, convert this energy to an equivalent in mass, use Einstein’s equation $E={\text{mc}}^2$. Hint: Divide the energy per m³ calculated in Exercise 29.27 by the speed of light squared. Check your units; you should have an answer in kg/m³. Now compare this answer with the critical density. Your answer should be several powers of 10 smaller than the critical density. In other words, you have found for yourself that the contribution of the CMB photons to the overall density of the universe is much, much smaller than the contribution made by stars and galaxies.

There is still some uncertainty in the Hubble constant. (a) Current estimates range from about 19.9 km/s per million light-years to 23 km/s per million light-years. Assume that the Hubble constant has been constant since the Big Bang. What is the possible range in the ages of the universe? Use the equation in the text, $T_0=\frac{1}{H},$ and make sure you use consistent units. (b) Twenty years ago, estimates for the Hubble constant ranged from 50 to 100 km/s per Mpc (which is 15–30 km/s per million light-years). What are the possible ages for the universe from those values? Can you rule out some of these possibilities on the basis of other evidence?

It is possible to derive the age of the universe given the value of the Hubble constant and the distance to a galaxy, again with the assumption that the value of the Hubble constant has not changed since the Big Bang. Consider a galaxy at a distance of 400 million light-years receding from us at a velocity, v. If the Hubble constant is 20 km/s per million light-years, what is its velocity? How long ago was that galaxy right next door to our own Galaxy if it has always been receding at its present rate? Express your answer in years. Since the universe began when all galaxies were very close together, this number is a rough estimate for the age of the universe.